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324                     ANALYTICAL MECHANICS
249. Logarithmic Decrement.  The logarii hm of the ratio of two consecutive amplitudes is constant and is called the logarithmic decrement of the motion. The amplitudes occur whenever the relation
CO*
tan (at) =  a
is satisfied. Let the first amplitude occur at the instant t = h; then since the period of the tangent is TT, the times of the succeeding amplitudes are given by
tan (wf) = tan (wti + HT) , u                               /     / _L 7l7r
or by                                   t  =  Cl + ~~ 
CO
where n is a positive integer. Hence, denoting the logarithmic decrement by X and the nth amplitude by anj we have
X = log 2-        (by definition)
a + 2
-        ^i        -    Sin^wfi 
"                   "         "'""
A6""flvl   " ......  v sin [wfe + (n + 2)
a(,rf^)
!       6         V         w/
= log - r-TTTT
2ir
~~~ 
0)
= ~P.                                              (XXV)
Therefore if I is known k" may be determined from observations of P and a.
* Obtained by setting^-  0.